Class 31

Pratice

Which image shows a positive relationship between the explanatory and response variables?

Practice

Which image shows a strong relationship between the explanatory and response variables?

Correlation

• Describes the direction and strength of the association between 2 numeric variables

Linear vs Non-Linear

Interpreting Correlations

Example: Poverty vs Graduation Rate

What’s the response variable?

Example: Poverty vs Graduation Rate

What’s the explanatory variable?

Example: Poverty vs Gradution Rate

Describe the relationship between these 2 variables.

Example: Poverty vs Graduation Rate

Example: Poverty vs Graduation Rate

Test Statistic

The test statistic \(t\) for the population Pearson correlation \(\rho\) (Greek letter rho) is estimated using the observed correlation \(r\).

\[ t=\frac{r\sqrt{n-2}}{\sqrt{1-r^2}} \]

Getting a p-value

Use the Student’s \(t\) distribution with degrees of freedom \(\text{df}=n-2\) to find a p-value for the observed correlation \(r\) in a sample of size \(n\) under the null hypothesis \(H_0 \colon \rho=0\).

# specify the test statistic and degrees of freedom
pt(test_statistic,
   df = n-2,
   lower.tail = F) # optional parameter

Eyeballing a Line

How do we find the best line to draw through variables that appear to have a linear relationship?

Quantifying Error: Residuals

Residuals are the difference between the observed values and the predicted values.

Example: Correlation vs Causation

Do ice cream sales cause drowning incidents?